Mathematics and Logic Behind Boolean and Fuzzy Computation

Abstract

Boolean algebra (BA) was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847) and set forth more fully in his An Investigation of the Laws of Thought (1854). According to Huntington, the term “Boolean algebra” was first suggested by Sheffer in 1913. Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1, considered to be “fuzzy.” By contrast, in Boolean logic, the truth values of variables may only be the “crisp” values 0 or 1. Fuzzy logic has been employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific (membership) functions. The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic had however been studied since the 1920s, as infinite-valued logic—notably by Lukasiewicz and Tarski. In this chapter we look at both of these logics holistically; any extensive details are beyond the scope of this book, and we encourage our readers to refer to so many books and articles that can be found on the Internet or Amazon.